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Modeling and Analysis of the Nonlinear Dynamics of aThermal Pulse Combustor

Subhashis Datta,* Achintya Mukhopadhyay† and Dipankar Sanyal‡

Jadavpur University, Kolkata 700032, India

Nonlinear dynamics of a thermal pulse combustor was investigated using a coupled fourth orderlumped combustor model. The effects of optically thin radiation from the combustor gases to thewall are also investigated. The system response changes from steady combustion to extinctionthrough oscillatory combustion as the wall temperature is lowered. The results suggest atransition to chaos through a period-doubling route prior to extinction. The system dynamics isqualitatively similar in presence and absence of radiation. However, inclusion of radiative heatloss leads to extinction at higher temperatures and also increases the predicted range of walltemperatures for which limit cycle behaviour is obtained. Results are also presented for theresponse of the system to sinusoidal perturbations in inlet mass flow rate. The response issensitive to both frequency and amplitude of the perturbation. The extinction is accelerated inpresence of a sinusoidal perturbation in inlet mass flowrate.

NomenclatureA Combustor surface area (m2)Ae Combustor cross-sectional area (m2)B Pre-exponential factor (m3/kgK0.5s)Cp Specific heat at constant pressure (J/kgK)DTP Diameter of tailpipe (m)f Friction factor (dimensionless)h Convective heat transfer coefficient (W/m2K)heff Effective heat transfer coefficient (W/m2K)Lc,1 First characteristic length (V/A)

(dimensionless)Lc,2 Second characteristic length (V/Ae)

(dimensionless)LTP Length of tailpipe (m)

im& Mass flow rate at combustor inlet (kg/s)

em& Mass flow rate at combustor exit (kg/s)

p Pressure (Pa)p0 Ambient pressure (Pa)

P p/p0 (dimensionless)

pe Pressure in tailpipe (bar)

eP pe/p0 (dimensionless)

T Temperature (K)Ta Activation temperature (K)T0 Ambient temperature (K)

T T/T0 (dimensionless)

* Research Scholar, Department of Mechanical Enginerring, Jadavpur University, Kolkata; Also: Lecturer,University Institute of Technology, The University of Burdwan, Burdwan.† Reader, Department of Mechanical Enginerring, Jadavpur University, Kolkata.‡ Professor, Department of Mechanical Enginerring, Jadavpur University, Kolkata.

42nd AIAA/ASME/SAE/ASEE Joint Propulsion Conference & Exhibit9 - 12 July 2006, Sacramento, California

AIAA 2006-4396

Copyright © 2006 by S. Datta, A. Mukhopadhyay and D. Sanyal. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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Te Temperature in tailpipe (K)

eT Te/T0 (dimensionless)

Tw Wall temperature (K)

wT Tw/T0 (dimensionless)

u Gas velocity in tailpipe (m/s)

u u/(Lc,2/τf) (dimensionless)

yf Fuel mass fraction (dimensionless)Zi /Vmi& (kg/m3s)

Ze /Vme& (kg/m3s)

ε Amplitude of mass flow rate oscillation(dimensionless)

γ Ratio of specific heats (dimensionless)κP Planck mean absorption coefficientρ0 Ambient density (kg/m3)σ Stefan-Boltzmann constant (W/m2K4)τc Characteristic flow time (s)τf Characteristic heat transfer time (s)τh Characteristic chemical reaction time (s)ω Angular frequency (rad/s)ωf Fuel consumption rate (kg/m3s)

I. Introductionulse combustors are known to have significantly higher thermal efficiencies, higher heat transfer rates and lowerpollutant emission than steady flow combustors. The operations of pulse combustors are characterized by self-sustained oscillations that have their origin in the strong coupling between the combustor dynamics and the flow

in the tailpipe (cf. Fig. 1). However, combustion with lean air-fuel ratios is susceptible to pressure oscillations andalso increased CO and hydrocarbon (HC) emissions. Stable combustion with optimal levels of NOx and CO and HCemission requires careful design. The operation and emission characteristics of the combustor are extremelysensitive to the design details and operating conditions of the combustor. Keller and Hongo [1] investigated variousfactors responsible for reduction in NOx production in a pulse combustor, compared to a steady combustor. Theyconcluded that lower NOx production was due to reduced residence time at peak temperature, caused by mixing ofthe combustor products with the cooler exhaust gases flowing back from the tailpipe. Keller et al. [2] examined theemission of CO and NOx from a lean premixed pulse combustor and concluded that the flame temperature, chemicalkinetics and residence time at high temperature were best controlled through equivalence ratio and macroscopicmixing. Thus the dynamics of the pulse combustor strongly influences the emission and performance of thecombustor.

Margolis [3] investigatedthe nonlinear dynamics associatedwith the thermoacoustic oscillationsin a model pulse combustor. Thenonlinear acoustic oscillations arisedue to combustion-driveninstabilities that excite one or moreacoustic modes. The study revealed

the existence of a limit cycle.Richards et al. [4] developed a perfectly-stirred reactor model for a thermal pulse combustor. The model

described the coupling between the combustor dynamics and flow in the tailpipe with a fourth order nonlinearlumped model. They showed that pulsating combustion could be obtained even with steady supplies of air and fuelwithout any mechanical valves. The results of the simulation compared favorably with those obtained from alaboratory-scale combustor.

P

InletTi,Yf,i Combustor Tailpipe

Fig. 1: Schematic of a Pulse Combustor

Te, Pe,Ue, Ae

T, P, Yf,A

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Daw et al. [5] demonstrated the existence of a bifurcation that ultimately leads to chaos throughinvestigations with a laboratory-scale pulse combustor and a model combustor, using the model of Ref. [4]. Thework used residence time as the bifurcation parameter. Rhode et al. [6] used the model of Richards et al. [4] todemonstrate that the flammable range of flow time by controlling the chaos using friction factor as the controlvariable. In et al. [7], on the other hand extended the flammable regime of a pulse combustor by maintaining (anti-control) chaos in the system that prevented the system from a transition to flame extinction. In their numericalexperiment, the only parameter varied was the residence time, which was used both as the bifurcating parameter andcontrol parameter. Edwards et al. [8, 9] used nonlinear feedback control to extend the operation of pulse combustorsto lean equivalence ratios and reduce levels of NOx and unburned hydrocarbon (UHC) emission by intermittentinjection of additional fuel.

Narayanaswami and Richards [10] developed the concept of pressure-gain combustion in gas turbinecombustors using pulsed combustors. They extended the model of Ref. [4] to include the expansion and backflow ofthe combustor gases into the inlet. In a companion paper, Richards and Gemmen [11] compared the modelpredictions and experimental observations and obtained qualitative agreement. They also used a scale analysis toobtain the dimensions of pressure-gain combustor.

Tang et al. [12] investigated the heat release timing in a nonpremixed pulse combustor. Their studiesindicted that the interaction between the complex flow and combustion processes causes the time delay needed to pr-oduce heat release oscillations that are nearly in phase with the pressure oscillations, thus assuring pulse combustionoperation. Kushari et al. [13] investigated the fuel effect on the dynamics of pulse combustors by comparing theamplitude and phase of oscillations for pulse combustors using methane, methane-carbondioxide and methane-

hydrogen-carbondioxide fuelblends. They found that additionof hydrogen extends the richflammability limit but decreasesthe amplitude of oscillations.

Marsano et al. [14]investigated the behaviour ofpulse combustors subjected tocyclic modulations of the massflowrate at inlet. Their workfocused on oscillations with largeamplitudes and frequencies closeto the natural frequency of thesystem. The results indicate that

the phase between the temperature and pressure traces is strongly affected by the oscillation characteristics.Kilicarslan [15 ] evaluated the frequency of a valved pulse combustor and compared the model predictions withexperiments.

Mukhopadhyay et al. [16] has proposed a novel technique for early detection of anomalies in a genericpulse combustor based on analysis of dynamic data from various sensors. The technique involves examining theresponse of the system to various known external stimuli in presence and absence of anomalies. The anomaly wasmodeled in the work through small changes in the value of friction factor (f). Significant variations in variousmeasures of anomaly were observed for very small changes in the value of f, even when the time series dataappeared nearly indistinguishable.

The above works did not consider the effect of parametric variations of combustor wall temperaturealthough results of several researchers indicated that the dynamics of the combustor is critically dependent on thewall temperature. Moreover, these models did not consider radiative heat loss to the walls from the gas, whichaccounts for a significant amount of heat loss. The objective of the present work is to develop a low dimensionalmodel of the pulse combustor that considers the effect of radiation and using the model to predict the effect of thewall temperature on the dynamic characteristics.

II. FormulationThe major assumptions of the present model are similar to that of Richards et al. [4]. However, radiation is

considered in the pulse combustor model. The major assumptions of the model are: (1) Perfectly-Stirred Reactor(PSR) for the combustor; (2) slug flow in the tailpipe; (3) constant specific heat and ideal gas model for the reactantand product gases; and (4) single-step Arrhenius model for chemical kinetics and (5) convective and radiative heat

Table – I: List of Model Parameters Used

A 0.0167 m2 LTP 0.61 mV 0.0001985 m3 p0 1 × 105 PaB 3.85 × 108 m3/kgK0.5s Ta 50Cp 1200 J/kgK T0 300 KDTP 0.0178 m Tw 1000 Kf 0.03 Yf,i 0.06h 120 W/m2K γ 1.27Lc,1 0.0119 m ρ0 1.12 kg/m3

Lc,2 0.7434 m τf 0.027 s

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loss to the wall. The model is described in terms of four ordinary differential equations, which are derived asfollows. The first equation is derived from a lumped model of conservation of energy within the reactor as

τγ+

ρ−γ+

τ−

τ

+τ

+τ

γ=hw0

e

f

i2

chf

ii

T

Z)1(

Z

P

T11TZ

P

T

dt

Td(1)

Combining Eq. (1) with conservation of mass gives the equation for pressure as

τ

+ρ

γ−

τ

+τ

+τ

γ=hw0

e

chf

ii

T

1ZT

11TZ

dt

Pd(2)

Conservation of fuel mass yields the equation for fuel mass fraction as

[ ]cc

0pfi,f

f

if 1

P

T

h

TCyy

1.

P

TZ

dt

dy

τ∆−−

τ= (3)

Finally, momentum balance in the tailpipe gives the equation for gas velocity in the tailpipe as

( )f

2,c

TP

3

ee

e

TP2,c

f0 L

D

1

u

u

2

f1P

P

T

LL

RT

dt

ud

τ−−

τ= (4)

In the above equations, the dimensionless characteristic times are defined as

i

0f Z

ρ=τ (5a)

weff

0p0ch Th

TCL1ρ

=τ (5b)

1

a2F2/3

2

0p

c*c T

Texpy

T

P

TC

hB−

−

∆′=τ (5c)

The chemical timescale, defined in Eq. (5c), is based on a single step global Arrhenius kinetics of the form

−ρ−=ω

T

TexpyyAT a

OF2

f2

1(6a)

For stoichiometric mixtures, using the relation that FOO yy ν= at all times, the above equation reduces to

−ρν−=ω

T

TexpyTA a2

F2

Of2

1(6b)

The above simplification enables us to eliminate the conservation equation for the oxidizer separately. The chemicaltimescale defined in Eq. (5c) assumes perfect mixing of the reactants. The parameter heff in Eq. (5b) reflects theeffective heat transfer coefficient that combines the convective and radiative heat losses. Radiation is modeled usingan optically thin gas approximation. With an optically thin model, the effective heat transfer coefficient can beexpressed as

( )( )TTTTLT4hh w22

wc30Peff 1

++σκ+= (7)

An expression for Ze needed to close the system is obtained from conservation of mass within the tailpipe as

e

e

fe T

puZ

τ= (8)

Finally, flow in the nozzle connecting the combustor and the tailpipe is assumed isentropic, owing to short length ofthe section, although irreversibilities are present in both the combustor and the tailpipe. Thus the pressure and thetemperature in the tailpipe are related to the combustor variables through isentropic relations as

2f0p

2c

2

eTC2

LuTT 2

τ−= (9a)

and)1(

T

Tpp e

e

−γγ

= (9b)

For externally forced systems, the mass flowrate at the inlet is perturbed sinusoidally as( )tSin1ZZ ii ωε+=′ (10)

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In the above equation, ε and ω denote the amplitude and angular frequency of perturbation respectively.The governing

equations are solved as asystem of couplednonlinear ordinarydifferential equationsusing the libraryfunction ODE45 ofcommercial package,MATLAB. Thisfunction uses a fourthorder Runge Kuttamethod. The time-stepsare selected adaptivelyby the integrator. In thepresent case, the upperbound of the time stepwas set at 10-5 seconds.However, the actualtime step used is muchfiner (~2.5×10-6 seconds). This gives adata sampling rate ofaround 40000 Hz. Thusthe generated dataset issufficiently large forpost-processing analysis.

III. Results and Discussions

A model pulsecombustor of specifiedgeometry wasinvestigated fordynamic response. Theparameters used in thecomputation areindicated in Table-I.Propane was used as thefuel and onlystoichiometric mixturesof fuel and air wereconsidered. The kineticparameters wereadopted from Ref. [4].To initiate the reaction,the initial temperaturewas raised to five timesthe ambienttemperature. Results arepresented here forparametric variation ofwall temperature.Variation of other

Fig. 2: Time series data at 1140 K

Fig. 3: Pressure plots for different wall temperatures

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parameters like frictionfactor and convectiveheat transfer coefficientproduced qualitativelysimilar results andhence are not presentedhere for brevity.

Initially theresults will be presentedconsidering radiativeloss from flames.Subsequently, theeffects of neglectingradiation. Figure 2shows the temporalvariations of all thestate variables, namely,temperature, pressure,fuel mass fraction andtailpipe velocity at awall temperature of1140 K. At this walltemperature, all thevariables undergo a

self-sustainedoscillation. The

mechanism driving this oscillation can be described as follows. Following the initial transients, combustion leads toan increase in pressure and temperature within the combustor and decrease in fuel mass fraction. The increasedpressure leads to an enhanced rate of mass efflux into the tailpipe. As the outflow rate exceeds the inflow rate, thepressure starts dropping. Similarly, increase in gas temperature increases the reaction rate, causing a fall in the valueof fuel mass fraction. This again eventually reduces the reaction rate, with a consequent decrease in gas temperature.As the pressure falls below the tailpipe pressure, a reversed flow starts taking place. Meanwhile, the fuelconcentration builds up due to continuous inflow of reacting mixture and reduced rate of fuel consumption due tolower temperature. This fuel buildup resumes the vigorous chemical reaction leading to repetition of the abovecycle. From measurement viewpoint, however, it is not easy to measure all the variables. Since the qualitativevariation is similar in all the four cases, results will be presented for pressure and, for phase plots, temperature only.

The time series data for pressure at various values of wall temperature are depicted in Figs. 3 and 4 toillustrate the evolution of the flame characteristics through the parameter space. In Fig. 3, the time series dataobtained by increasing the wall temperature from 1140 K are presented. As the wall temperature increases to 1158K, the amplitude of the oscillation increases with time, typical of an unstable configuration. However, with a slightincrease in the wall temperature (1165 K), the nature of the oscillation changes completely and the oscillationbecomes damped. The damping is even more pronounced at 1170 K. At 1170 K, the oscillation is characterized by adamped oscillation with very small amplitude.

The temporal variation of the combustor pressure as the wall temperature is decreased from 1140 K ispresented in Fig. 4. As the wall temperature decreases to 1090 K, a lower value of peak pressure appears alternatelywith the higher value peak, at regular intervals. Thus the effective time period between two similar peaks becometwice the original time period. This is a characteristic feature of period-doubling bifurcation and is a possible routeto chaos [17]. As the temperature decreases further, additional lower values of peaks appear and the combustionbecomes irregular. At 1040 K, the combustion is highly erratic and intermittent. As the wall temperature decreases,the heat loss to the wall increases and this tends to promote extinction of the flame. However, since the inflow issteady, weakened combustion leads to accumulation of fuel in the combustor. The increased fuel concentrationoffsets the effect of decreasing gas temperature and the flame is reestablished. The drastic change in combustioncharacteristics due to small changes in wall temperature clearly bring out the strong nonlinear dependence of thereaction rate on temperature. A further decrease of wall temperature leads to complete extinction.

Fig. 4: Pressure plots for different wall temperatures

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The time series data presented above is indicative of a possible transition to chaos. However, the nature ofsuch variation is better represented in terms of phase plots. The phase plots in Fig. 5 are obtained by projecting thetrajectory on to pressure-temperature plane. At 1158 K, the phase plot shows an expanding spiral. As the walltemperature decreases to 1140 K, the nature of the plot changes to a limit cycle. This corroborates the periodicnature of the pressure trace, apparent from the time series data. As the wall temperature decreases further, theexistence of a period-doubling is clearly illustrated by the phase plot at 1090 K. Although phase plots are non-intersecting, the apparent intersection of the phase plots for lower values of wall temperature (1090 K and below)can be explained by the fact that these are actually projections of the phase plot on a two-dimensional (P-T) plane.The intersecting nature of the projections suggests that for these values of wall temperatures, the effective dimension

of the system exceedstwo. This is a necessarycondition for transitionto chaos. As the walltemperature decreases,increasing complexityof the phase plotindicates that thesystem undergoesfurther bifurcations.The extremely complexshapes of the phaseplots at 1050 K and1040 K indicate apossible transition tochaos. The inner curvesat 1040 K correspond tonear extinction stateswhen the dimensionlesspressure stays close tounity.

One difficultyof constructing phaseplots for systems withhigh dimension is therequirement of differenttypes of sensors forrecording the timeseries data of differentvariables [17]. Astrategy commonlyadopted in dynamicanalysis to overcomethis difficulty involvesuse of TakensEmbedding Theorem

[18]. The basic premise of this theorem is that the dynamics of a multidimensional system can be reconstructed fromthe time series data of a single variable. However, to achieve this, the multidimensional behaviour of the system hasto be reconstructed by constructing a vector consisting of the values of a single variable at different time instants,rather than using different variables at a single instant of time.

A crucial parameter for this reconstruction is a proper choice of delay time, that is, the time gap twosuccessive elements of the vector. A very small value of the delay time may not reveal all the details of the dynamiccharacteristics, while a very large value may introduce spurious features. Figure 6 shows the reconstructed phaseplots for different delay times for two conditions. The upper plots correspond to the limit cycle oscillation at 1140 K,while the lower one refers to the intermittent combustion, close to extinction, at 1040 K.

Fig. 5: Phase Plots for Different Wall Temperatures

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For the limit cyclecorresponding to 1140 K, as the delaytime increases, the shape of theattractor shows remarkable changes.For a delay of 200 timesteps, the shapeof the attractor indicates that thepressure values at the two instantsincrease and decrease in tandem. Thisindicates a small difference in phasedifference. On the other hand, as thetime delay increases to 700 steps, thephase difference between the twopressures increases sufficiently suchthat for a significant portion of thecycle, the two pressure values showopposite tends. On further increase ofthe delay to 1500 timesteps, the twopressure values are never in tandem,indicating a phase difference of over90 degrees. However, at a delay of3000 steps, the attractor stronglyresembles the attractor at a delay of200 timesteps, indicating completion ofa time period. In fact, from the timeseries data, a time period of about 7 msis observed. For an average timestep of2.5×10-6s, a complete cyclecorresponds to about 2800 timesteps.Thus the reconstructed phase plotscorrectly reproduce the systemdynamics.

At 1040 K also, the delayplots show a similar sensitivity to timedelay. An inspection of the time seriesdata reveals that the cycle frequenciesdo not appreciably change with walltemperature. However, at lower walltemperatures, other low frequency andintermittent events also occur, givingthe system dynamics a complexcharacter. Thus the basiccharacteristics of the delay plots aresimilar for 1140 K and 1040 K.However, the presence of additionalevents renders the attractors complex(and possibly chaotic) character. The

most significant observation, however, is that the system dynamics at a delay of 3000 steps, does not return to thedelay plot with 200 timesteps. This indicates the presence of large cycle-to-cycle variations.

Figure 7 presents the reconstructed phase plots or return maps for the cases shown in Fig. 5. The utility ofthis method of representation and appropriateness of the delay time is evident from the fact that Fig. 7 revealsfeatures, qualitatively similar to those of Fig. 5. The limit cycle behaviour at 1140 K, the period doubling at 1090 Kand irregular and intermittent combustion at 1040 K are all clearly revealed.

Fig. 6: Effect of Lag Time on Return Maps for Different WallTemperatures

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The transitionin the dynamiccharacteristics of thepulse combustor issummarized in Fig. 8. InFig. 8, the local peakvalues of pressure areplotted for different walltemperatures. For limitcycle behaviour, thesame peak value isachieved in all thecycles. Consequently,for such cases, only onepeak pressure isobserved. On the otherhand, after a perioddoubling bifurcation, ahigh peak alternates witha low peak, which givesrise to two peaks.Likewise, as thecombustion patternbecomes more irregular,the number of peakvalues keeps onincreasing. From thefigure, it is observed thatthe system possesses alimit cycle in thetemperature range 1140K to 1090 K. Beyond1140 K, the pressurefluctuation occurs with aregular frequency butincreasing amplitude.This leads toprogressively largernumber of peaks as the

wall temperature increases. However, beyond approximately 1160 K, the trend reverses and the number of peaksstarts decreasing and finally falls to a single low value at 1170 K. This decreasing number of peaks corresponds tothe damped oscillation observed earlier. From the figure, it is clear that as the wall temperature is increased from1140 K, the system undergoes a transition to a steady state that attains a low pressure. As the temperature is

decreased below 1140K, the number of peaksincreases as the systemundergoes successiveperiod doublingtransitions. Finally atabout 1050 K, thecombustion becomesirregular and erratic,corroborated by a largenumber of peaks.Beyond 1025 K, the

Fig. 7: Return Maps for Different Wall Temperatures

Fig.8: Effect of Radiation on Combustor Dynamics

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flame extinguishes permanently.The lower part of Fig. 8 generates a similar bifurcation plot but using a model that neglects radiative heat loss to

the wall. The qualitative pattern of the system transition is similar to that of the earlier case, but the transitions occurat lower temperatures. This is expected, as neglecting radiation reduces the effective heat transfer coefficient and toobserve a similar effect, the temperature difference has to be increased. Another significant observation is that thepredicted range of limit cycle operation increases considerably when radiative losses are taken into account.Radiation introduces additional nonlinearities in the model. This promotes the attainment of limit cycles, which is anonlinear event.

Figure 9 shows thetime series data of thesystem response to asinusoidal perturbationin mass flow rate. Theamplitude ofperturbation is 2% ofthe mean mass flowrateand the frequency ofoscillation is 10 Hz. Thewall temperature isprogressively loweredfrom 1200 K tillextinction of the flame.At a wall of temperatureof 1200 K (not shown),the combustor pressurereaches a steady value,following the initialtransience. As the walltemperature is lowered,low amplitudeoscillations start toappear, with theamplitude increasing asthe temperature islowered. However, asthe wall temperature islowered from 1170 K to1160 K, the dynamics ofthe system changesfrom damped oscillationto unstable oscillations.In each of these cases,however, a lowfrequency oscillation,characteristic of theexternal perturbation, issuperposed on the highfrequency self-sustainedoscillations. As thetemperature is further

lowered to below 1150 K (shown in Fig. 2 for 1140K), a large amplitude high frequency oscillation appearssuperposed with the low frequency forced oscillation. As the temperature is further reduced to 1090 K, pressurepulses with lower peak values appear at regular intervals. For unforced systems, this temperature range correspondsto a period-doubling bifurcation. However, the period-doubling effect is not so pronounced in this case due to thepresence of the low frequency oscillation. On the contrary, the lower peaks appear pronounced when they coincidewith the trough of the low frequency wave and are almost suppressed near the crests of the low frequency waves. On

Fig. 9: Time Series Data for Combustor Pressure for ε = 0.02 and Frequency = 10 Hz

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further reduction of thewall temperature, thecombustion becomesextremely irregular at1070 K, withoccasional extinctionand reignition. Furtherlowering of the walltemperature to about1060 K leads toextinction.

Figure 10 shows thepressure variations fora stimulus in mass flowrate with ε = 0.05 andfrequency of 10 Hz. Acomparison with Fig. 2clearly reveals theeffect of amplitude ofthe mass flow rateoscillation on thedynamics of thesystem. At high walltemperatures thebehaviour isqualitatively similar.However, theamplitude of the lowfrequency pressureoscillations ispredictably higher. Thequalitative patterns forthe two cases aresimilar for the regimeof steady oscillation.However, at higheramplitude, transitionfrom steady oscillationto extinction is suddenas is evident from thetransition from 1106 Kto 1104 K in Fig.2.

This sudden transition results in extinction at a much higher temperature.Figure 11 shows the pressure variations for a disturbance with 50 Hz frequency and a dimensionless amplitude

of 0.05. A comparison with Fig. 9 reveals the effect of the forcing frequency. It is observed that the pressure tracesshow some irregular features even at high wall temperatures. However, the most notable feature is that the steadyoscillation is not achieved in this case. Instead, the system undergoes a transition to extinction even at high walltemperatures. A significant observation from the results is that the system behaviour is significantly affected,particularly close to extinction, even when the forcing frequency is much lower than the dominant natural frequency.This can be explained by the fact that close to extinction, the oscillations become irregular and several lowerfrequency modes are observed. There is strong nonlinear coupling between these modes and the forced oscillationdue to sinusoidal variation in mass flowrate at inlet.

Fig. 10: Time Series Data for Combustor Pressure for ε = 0.05 and Frequency = 10 Hz

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IV. ConclusionsA fourth order nonlinear model has been developed for investigating the dynamics of a thermal pulse

combustor. The model employs well-stirred reactor model and single step Arrhenius chemistry. The effects ofoptically thin radiation and unmodeled dynamics due to improper mixing are also investigated. The systemdynamics is qualitatively similar in presence and absence of radiation. However, inclusion of radiative heat lossleads to extinction at higher temperatures.

Fig. 11: Time Series Data for Combustor Pressure for ε = 0.05 and Frequency = 50 Hz

The system exhibits a steady oscillation at a wall temperature of 1140 K. As the wall temperature is increased,the oscillation first becomes unstable up to a wall temperature of around 1165 K and then gets damped and finallyattains a non-oscillating steady character. As the wall temperature decreases below 1140 K, the system undergoes aperiod doubling bifurcation and ultimately at low wall temperatures, the combustion becomes irregular andintermittent. Finally, below 1025 K, the flame extinguishes. The phase plots and return maps indicate a possibletransition to chaos before extinction.

The model has also been used to investigate the effect of sinusoidal perturbations in mass flow rate. Theresponse of the system strongly depends on the amplitude and frequency of the perturbation, even for lowamplitudes and frequencies significantly lower than the natural frequency of the system. For the range of amplitudesand frequencies of the external stimulus investigated, it is found that increase in both amplitude and frequencypromotes early extinction.

V. References

[1] Keller, J.O. and Hongo, I., Pulse Combustion: The Mechanism of NOx Production, Combust. Flame, 80, 219 –237, 1990.

[2] Keller, J.O., Bramlette, T.T., Barr, P.K. and Alvarez, J.R., NOx and CO Emissions from a Pulse CombustorOperating in a Lean Premixed Mode, Combust. Flame, 99, 460 – 466, 1994.

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[3] Margolis, S.B., The Nonlinear Dynamics of Intrinsic Acoustic Oscillations in a Model Pulse Combustor,Combust. Flame, 99, 311 – 322, 1994.

[4] Richards, G.A., Morris, G.J., Shaw, D.W., Kelley, S.A. and Welter, M.J., Thermal Pulse Combustion, Combust.Sci. Technol., 94, 57 – 85, 1993.

[5] Daw, C. S., Thomas, J. F., Richards, G. A. and Narayanaswamy, L. L., Chaos in Thermal Pulse Combustion,Chaos, 5 pp. 662 – 670, 1995.

[6] Rhode, M. A., Rollins, R. W., Markworth, A. J., Edwards, K. D., Nguyen, K., Daw, C. S. and Thomas, J. F.,Controlling Chaos in a Model of Thermal Pulse Combustion, J. App. Phys., 78, 2224 – 2232, 1995.

[7] In, V., Spano, M. L., Neff, J. D., Ditto, W. L., Daw, C. S., Edwards, K.D. and Nguyen, K., Maintenance ofChaos in a Computational Model of Thermal Pulse Combustor, Chaos, 7, 605 – 613, 1997.

[8] Edwards, K. D., Finney, C. E. A., Nguyen, K. and Daw, C. S., Application of Nonlinear Feedback Control toEnhance the Operation of a Pulsed Combustor, Proceedings of the 2000 Spring Technical Meeting of the CentralStates Sections of the Combustion Institute, 2000.

[9] Edwards, K. D., Nguyen, K. and Daw, C. S., Enhancing the Operation of a Pulsed Combustor with Trajectory-Correction Control, Proceedings of the Second Joint Meeting of the U.S. Sections of the Combustion Institute,2001.

[10] Narayanaswami, L. and Richards, G.A., Pressure-Gain Combustion: Part I – Model Development, Trans.ASME J. Engg. Gas Turbines Power, 118, 461 – 468, 1996.

[11] Richards, G.A. and Gemmen, R.S., Pressure-Gain Combustion: Part II – Experimental and Model Results,Trans. ASME J. Engg. Gas Turbines Power, 118, 469 – 473, 1996.

[12] Tang, Y.M., Waldherr, G., Jagoda, J.I. and Zinn, B.T., Heat Release Timing in a Nonpremixed Helmholtz PulseCombustor, Combust. Flame, 100, 251 – 261, 1995.

[13] Kushari, A., Rosen, L.J., Jagoda, J.I. and Zinn, B.T., The Effect of Heat Content and Composition of Fuel onPulse Combustor Performance, Proc. Combust. Inst., 26, 3363 – 3368, 1996.

[14] Marsano, S., Bowen, P.J. and O’Doherty, T., Cyclic Modulation Characteristics of Pulse Combustors, Proc.Combust. Inst., 27, 3155 – 3162, 1998.

[15] Kilicarslan, A., Frequency Evaluation of a Gas-Fired Pulse Combustor, Int. J. Energy Research, 29, 439 – 454,2005.

[16] Mukhopadhyay, A., Datta, S., Gupta, S., Ray, A. and Yang, V., Anomaly Detection in a Generic Thermal PulseCombustor, 2003 Fall Technical Meeting of the Eastern States Section of the Combustion Institute, October2003.

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